Gender Inequality and Economic Growth in Korea

Jinyoung Kim, Jong-Wha Lee and Kwanho Shin Korea University

Abstract

Calibrating an endogenous-growth model to Korean data, we analyze the impact of gender inequality in Korea on long-term economic growth. We find that gender equality policies that lower discrimination in the labor market or that increase the time spent by a father on child-rearing can contribute positively to female labor market participation and per capita income growth. The simulation results show that when the disparities between men and women at home and in the labor market are completely removed, the female labor force participation rate increases from 54.4% to 67.5%, and the growth rate in per capita income rises from 3.6% to 4.1% on average over a generation. JEL codes: O53, E24, J13, J71 Keywords: gender inequality, economic growth, female labor market participation, human capital accumulation, Korea

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Jinyoung Kim: jinykim@korea.ac.kr. This research was carried out as part of the gender equality and development project funded by the Asian Development Bank. The earlier version of this paper appeared as ADB Working Paper No. 473 titled “Impact of Gender Inequality on the Republic of Korea’s Long-term Economic Growth: An Application of the Theoretical Model of Gender Inequality and Economic Growth.”

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1. Introduction

South Korea has made significant economic progress in the past 50 years, as

demonstrated by the increase in its per capita income from just $80 in 1960 to over $24,000

in 2013. A critical factor for Korea’s economic success has been its fast-growing, well-

educated labor force. From 1960 to 2010 the share of adults who had completed secondary

schooling or higher soared from 20% to an impressive 87% in 2010 (Barro and Lee, 2013).

This abundance of well-educated workers has brought about higher levels of labor

productivity and higher returns on investment and has developed capabilities for facilitating

technological adoption and innovation. Low-cost and good-quality labor thus became the

foundation for Korea’s successful export-oriented development strategy.

During its period of rapid industrialization and development, Korea made substantial

strides toward gender equality in education and employment opportunities. The gender gap

has become negligible in the secondary school enrollment rates and in the advancement rates

to higher education.1 More noticeable than ever before is the presence of women in such

elite professions as law, medicine, and high-level civil service. The general improvement in

gender equality in Korea is also manifest in the change in sex ratio at birth.2

However, there is still a significant gender gap in labor market participation in Korea.

According to the OECD data in 2012, only 55% of Korean women from ages 15 to 64 are in

the labor force compared to 65% for OECD countries on average. Korea’s female labor force

participation rate (LFPR) substantially lags behind the male counterpart: the difference in

1 In 1990, the average years of schooling for Korean women in their 30s were 10.4 while the average years for men were 11.8. By 2010, those averages had risen to 13.9 for women and to 14.1 for men. However, if we consider only women in their 20s, their average schooling years actually reached 14.3 in 2010, even higher than the 13.9 years of their male counterparts (Social Indicators in Korea, Statistics Korea).

2 In 1990, at the peak of its rise, the sex ratio at birth—measured by the number of boys born per 100 girls—reached 116.5. By 2007, that ratio fell down to the normal level of 106.

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LFPR between men and women was 23.4 percentage points in Korea against 17.5 percentage

points on average in all the OECD countries in 2011. This wide gender gap in LFPR in Korea

has been quite persistent over time.

The labor force participation rate of Korean women shows an M-shaped pattern over

the life cycle. Due to career interruption after marriage or child birth, the rate drops

significantly in their late 20s and early 30s (see Figure 1).3 Many Korean women in their

30’s are eventually recruited back to the labor market but child rearing remains a major

obstacle for highly educated female workers who want to continue their careers.4 At all

levels of incomes, Korean mothers remain primarily responsible for housework and childcare,

and inflexible work environments along with a lack of affordable, good-quality childcare

facilities make it challenging for them to balance work and home.

Besides market participation, Korean women are also experiencing gender inequality

in various other parts of the society. Table 1 shows various measures of gender gap between

men and women in Korea. Korean women are reported to have lower employment rates

(especially among those highly educated), be less likely to own their businesses, earn less in

wage, and be less represented and participating in politics, government, and corporate boards

than men.

Korean women generally have a strong perception that there is gender inequality in

various parts of society. According to the 2002 Social Survey by Statistics Korea, 72.4% of

women have that perception. This finding was also borne out by the global gender gap report

3 Between Korea’s married men and women, the gender gap in LFPR is much wider. In 2011, the LFPR of married men in Korea was at 82.8% and that of single men at 52.2%. On the other hand, the LFPR of married women was at 49.3% and that of single women at 50.9%.

4 The Korea Time Use Survey conducted in 2009 reports that time spent by a wife for childcare is, on average, more than 3 times longer than that by a husband—55 minutes per day vs. 14 minutes.

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of the World Economic Forum, which ranked Korea 111th in gender equality in 2013 (Table

2). The WEF index takes into account women’s general standing in (i) economic participation

and opportunity, (ii) educational attainment, (iii) health and (iv) political empowerment.

Through a three-year plan for economic innovation announced in February 2014, the

Korean government is currently trying to address the gender inequality issue head-on. The

major goal of this plan is to increase the female employment rate from the current 54% to 62%

by 2017. This goal will be pursued by encouraging female workers to remain in the

workforce through such measures as providing affordable, good-quality childcare facilities

and expanding paid parental leave (see detailed description of the Korean government’s

policies on gender inequality in Appendix A).

How would this gender-based government policy be effective? Which policy would

be most effective in enhancing the growth rate of per capita income? In this paper we attempt

to address these questions by quantifying the effects of various policies on female labor force

participation and on Korea’s long-term economic growth. Another objective of this paper is

to assess the cost of gender inequality in term of output lost due to the practice of gender

inequality, and examine how high the female labor force participation rate can rise if gender

bias is eliminated.

The theoretical framework in this paper is largely an extension of our model in Kim,

Lee and Shin (2016) which accounts endogenously for women’s time allocation between

home production, child rearing, and market work. We extend the model by introducing

various aspects that are relevant to the Korean economy in particular. We calibrate the

extended model to fit its steady-state values to the observed values from Korea and conduct

simulations to quantitatively measure the opportunity cost of gender inequality in terms of

foregone output as well as the impacts of gender-based policies on women’s labor market

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participation and economic growth.

There is an increasing body of literature on gender equality and growth.5 Existing

theoretical literature emphasizes three channels through which gender equality influences

growth— female labor market participation, average human capital stock, and fertility. A

considerable number of empirical papers have investigated the impact of gender inequality in

education and employment on economic growth and the majority of these studies have found

that gender inequality indeed adversely affects economic growth.

As gender inequality has lately become a pressing issue in Korean society, many

Korean researchers have studied various aspects of female labor supply and household work,

mostly using microeconomic perspectives. Kim and Sung (2007), Woo (2008), Cho (2009)

and Choi (2011) have estimated the labor supply function of Korean women empirically or

by model calibration in their investigation of the effects of various government policies like

subsidies for childcare and earned income tax credits. Kim and Cho (2003) and Kim (2012)

have studied the determinants of labor market reentry by married women after childbirth or

childcare leave, including several gender-related policies. Huh (2008) has examined the

factors for time spent on household production by men and women.

To the best of our knowledge, no academic research using macroeconomic

perspectives has yet been undertaken to assess the effects of gender inequality on economic

growth in the Korean economy and this paper is an effort to fill this gap. The paper proceeds

as follows after this introductory section: In Section 2, we introduce the formal endogenous-

growth model considering various aspects of gender disparity. We then calibrate the model

and derive the benchmark steady state that is characterized by a balanced growth path. In

5 See Kim et al. (2016) for a succinct survey of theoretical papers on this topic, including Galor and Weil (1996) and Lagerlof (2003).

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Section 3, we experiment on the effects of gender equality policies and estimate the output

cost of gender inequality. Section 4 provides concluding observations.

2. The Theoretical Model

Our theoretical model is built on Agenor (2016), Agenor and Canuto (2015) and Kim

et al. (2016). The model has a three-period overlapping generations (OLG) structure where

various aspects of gender inequality are related to the economy’s growth performance. Based

on a similar model, Kim et al. (2016) have shown that improving gender equality can

contribute significantly to economic growth by changing female time allocation and

promoting accumulation of human capital.6

In order to fit the model better to the Korean economy, we modify Kim et al (2016) in

several directions. First, aside from market and home production, males are assumed to

allocate some of their time to child rearing and education. This modification is essential for

examining the impact of a government policy that encourages males to spend more time in

child rearing.

Second, we assume that the father’s education level as well as that of the mother’s

determines the accumulation of human capital by their children. This change, along with the

change in the first modification that males allocate time to education, will allow us to

investigate the perfect gender-equality case where males and females behave exactly in the

same way.

6 The model in Kim et al. (2016) simplifies Agénor (2016), but differs in several important dimensions. First, Kim et al. (2016) explicitly considers the difference between the quantity and quality of children in terms of their costs, following Becker et al. (1990), and allows the altruism in utility, as in Ehrlich et al. (1991). Second, the model assumes the existence of a fixed cost per child and a distinct time cost in educating children. Finally, it also assumes that husbands take limited responsibility in household production, which is one aspect of the gender gap.

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Third, we remove the parental bias in favor of sons in time allocation and in the

preference for educating children because there seems to be no evidence now of that bias in

favor of sons.

Finally, we assume in the benchmark model that the government revenue is not spent

on any growth-enhancing activities. This will allow us to explore a possible government

policy that switches spending from unproductive usage to education.

2.1 Model Structure

Each individual’s lifetime has three periods: childhood, adulthood (middle age) and

retirement (old age). There is a continuum of identical families consisting of parents born at

time (t-1) and children born at time t. The family’s utility function at time t is as follows:

𝑈𝑈𝑡𝑡 = 𝜂𝜂𝑐𝑐1

1 − 𝜎𝜎𝑐𝑐𝑡𝑡1−𝜎𝜎 + 𝜂𝜂𝑞𝑞

11 − 𝜎𝜎

𝑞𝑞𝑡𝑡1−𝜎𝜎

+𝜂𝜂𝑒𝑒 �1

1−𝜎𝜎��𝑛𝑛𝑡𝑡

2�𝛿𝛿𝑒𝑒𝑡𝑡+1𝑚𝑚 �

1−𝜎𝜎+ 1

1−𝜎𝜎��𝑛𝑛𝑡𝑡

2�𝛿𝛿𝑒𝑒𝑡𝑡+1𝑓𝑓 �

1−𝜎𝜎� + 𝑝𝑝𝐴𝐴

1+𝜌𝜌1

1−𝜎𝜎𝑐𝑐𝑡𝑡+11−𝜎𝜎 (1)

where 𝑐𝑐𝑡𝑡 (𝑐𝑐𝑡𝑡+1) is the family’s total consumption during the parents’ adulthood (parents’

retirement), 𝑞𝑞𝑡𝑡 consumption (and production) of home goods, 𝑛𝑛𝑡𝑡 the number of children

(of which half are sons and the other half are daughters), 𝑒𝑒𝑡𝑡+1𝑚𝑚 (𝑒𝑒𝑡𝑡+1𝑓𝑓 ) the education level of

sons (daughters) that will determine the efficiency of male (female) adult workers at t+1,

𝜌𝜌 > 0 the time discount rate, 𝜎𝜎−1 the intertemporal elasticity of substitution, and 𝑝𝑝𝐴𝐴 the

probability of survival from adulthood to retirement. The coefficient cη pertains to relative

preference for today’s consumption, 𝜂𝜂𝑞𝑞 to relative preference for the home-produced good,

and 𝜂𝜂𝑒𝑒 to relative preference for children’s education.7

7 In our model, an agent in retirement does not spend time on market production, child education, or child rearing, and therefore obtains utility only from consumption.

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We assume that the female adult divides her time for four uses: market production,

home production, child rearing, and child education. Thus the time constraint for the female

is as follows:

ℎ𝑡𝑡𝑤𝑤 + ℎ𝑡𝑡𝑞𝑞 + ℎ𝑡𝑡𝑅𝑅 + ℎ𝑡𝑡𝑒𝑒 = 1 (2)

where ℎ𝑡𝑡𝑤𝑤 is the adult female’s time allocated to market production, ℎ𝑡𝑡𝑞𝑞 her time allocated

to home production, ℎ𝑡𝑡𝑅𝑅 her time allocated to child rearing, and ℎ𝑡𝑡 𝑒𝑒 her time allocated to

child education.

We assume that

ℎ𝑡𝑡𝑚𝑚𝑞𝑞 = 𝑓𝑓1ℎ𝑡𝑡

𝑞𝑞 (3)

where ℎ𝑡𝑡𝑚𝑚𝑞𝑞 is the male adult’s time allocated to home production and 𝑓𝑓1 represents the

bargaining power of a wife with respect to home production. Equation 3 implies that the

decision on time allocation to home production by the male and by the female is done in two

steps: first, the decision on the female’s time allocated to home production is made, and then

second, the bargaining power of a female determines the male’s proportionate time allocation

to home production. The bigger the bargaining power of the women, the higher is the

proportion. For simplicity, we assume that 𝑓𝑓1 is exogenously determined and constant.

Generally 𝑓𝑓1 < 1 and perfect equality is obtained if 𝑓𝑓1 = 1. This two-step decision allows

us to focus solely on the female’s decision, thus simplifying the problem. We will make the

same assumption for the time allocation to other uses.

We assume that ℎ𝑡𝑡𝑅𝑅 = (2 − 𝑓𝑓2)𝑣𝑣𝑛𝑛𝑡𝑡 where 2𝑣𝑣 is total rearing time needed per child

as a mother and a father each spends 𝑣𝑣. Again the male’s time allocated to child rearing is

ℎ𝑡𝑡𝑚𝑚𝑅𝑅 = 𝑓𝑓2𝑣𝑣𝑛𝑛𝑡𝑡 (4)

where 𝑓𝑓2 represents the bargaining power of a female with respect to child rearing. 𝑓𝑓2 is not

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necessarily equal to 𝑓𝑓1 since the comparative advantage of the male and the female in these

activities are not the same. In general, the female has more comparative advantage in child

rearing particularly if child rearing also involves breast feeding.

Finally the time allocated to education satisfies ℎ𝑡𝑡𝑒𝑒 = 𝑛𝑛𝑡𝑡𝜖𝜖𝑡𝑡𝑒𝑒 where 𝜖𝜖𝑡𝑡𝑒𝑒 is average

education time spent for each child. We assume that the female allocates her time equally

between sons and daughters. The male’s time allocated to child rearing per child is

determined by

𝜖𝜖𝑡𝑡𝑚𝑚𝑒𝑒 = 𝑓𝑓3𝜖𝜖𝑡𝑡𝑒𝑒 (5)

where 𝑓𝑓3 represents the bargaining power of a female with respect to child education. Hence

the total time spent on child education by the male is ℎ𝑡𝑡𝑚𝑚𝑒𝑒 = 𝑓𝑓3𝑛𝑛𝑡𝑡𝜖𝜖𝑡𝑡𝑒𝑒.

Then the time constraint faced by the female can be represented as follows:

ℎ𝑡𝑡𝑤𝑤 + ℎ𝑡𝑡𝑞𝑞 + (2 − 𝑓𝑓2)𝑣𝑣𝑛𝑛𝑡𝑡 + 𝑛𝑛𝑡𝑡𝜖𝜖𝑡𝑡𝑒𝑒 = 1 (6)

The home production function is

𝑞𝑞𝑡𝑡 = 𝑞𝑞�(ℎ𝑡𝑡𝑞𝑞 + ℎ𝑡𝑡

𝑚𝑚𝑞𝑞)𝛾𝛾[�𝑒𝑒𝑡𝑡𝑓𝑓�

𝜒𝜒(𝑒𝑒𝑡𝑡𝑚𝑚)1−𝜒𝜒]= 𝑞𝑞�(1 + 𝑓𝑓1)𝛾𝛾(ℎ𝑡𝑡

𝑞𝑞)𝛾𝛾[�𝑒𝑒𝑡𝑡𝑓𝑓�

𝜒𝜒(𝑒𝑒𝑡𝑡𝑚𝑚)1−𝜒𝜒] (7)

where 𝑒𝑒𝑡𝑡𝑓𝑓 and 𝑒𝑒𝑡𝑡𝑚𝑚 are the education level of mother and father, and the second equality

holds because of (3). We assume that time spent by a male is perfectly substitutable with time

spent by a female. However, we assume that the education of a female and a male is

introduced as a Cobb-Douglas functional form where 𝜒𝜒 and 1 − 𝜒𝜒 are the output elasticity

of female and male education.

The education level of children that will become productivity when they become

adults is determined by three factors: the average government spending on education per

(surviving) child, a mother’s human capital 𝑒𝑒𝑡𝑡𝑓𝑓, and the time mothers allocate to each child,

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as follows:8

𝑒𝑒𝑡𝑡+1𝑓𝑓 = 𝑒𝑒𝑡𝑡+1𝑚𝑚 = �̅�𝑒( 𝜇𝜇𝐺𝐺𝑡𝑡

𝑛𝑛𝑡𝑡𝑎𝑎𝑁𝑁𝑡𝑡/2

)𝜈𝜈1[�𝑒𝑒𝑡𝑡𝑓𝑓�

𝜒𝜒1(𝑒𝑒𝑡𝑡𝑚𝑚)1−𝜒𝜒1]1−𝜈𝜈1((1 + 𝑓𝑓3)𝜖𝜖𝑡𝑡𝑒𝑒)𝜈𝜈2 (8)

where 𝐺𝐺𝑡𝑡 is total government spending, 𝜇𝜇 an indicator of efficiency of government

spending, 𝑁𝑁𝑡𝑡 the number of individuals of generation t, and 𝑛𝑛𝑡𝑡𝑎𝑎 the average number of

children in the households. Since we assume the representative household, 𝑛𝑛𝑡𝑡𝑎𝑎 = 𝑛𝑛𝑡𝑡 holds in

equilibrium.

The household budget constraint at t and t+1 are9:

𝑐𝑐𝑡𝑡 + 𝑠𝑠𝑡𝑡 = (1 − 𝜏𝜏)𝑤𝑤𝑡𝑡𝐻𝐻 (9)

𝑐𝑐𝑡𝑡+1 = (1+𝑟𝑟𝑡𝑡+1)𝑠𝑠𝑡𝑡𝑝𝑝𝐴𝐴

(10)

where τ ∈ (0,1) is the tax rate, 𝑠𝑠𝑡𝑡 saving, 𝑟𝑟𝑡𝑡+1 interest rate between t and t+1, and 𝑤𝑤𝑡𝑡𝐻𝐻

total gross wage income for the household.

𝑤𝑤𝑡𝑡𝐻𝐻 = 𝑒𝑒𝑡𝑡𝑚𝑚ℎ𝑡𝑡𝑚𝑚𝑤𝑤𝑤𝑤𝑡𝑡

𝑚𝑚 + 𝑒𝑒𝑡𝑡𝑓𝑓ℎ𝑡𝑡𝑤𝑤𝑤𝑤𝑡𝑡

𝑓𝑓

= 𝑒𝑒𝑡𝑡𝑚𝑚(1 − 𝑓𝑓1ℎ𝑡𝑡𝑞𝑞 − 𝑓𝑓2𝑣𝑣𝑛𝑛𝑡𝑡 − 𝑓𝑓3𝑛𝑛𝑡𝑡𝜖𝜖𝑡𝑡𝑒𝑒)𝑤𝑤𝑡𝑡𝑚𝑚 + 𝑒𝑒𝑡𝑡

𝑓𝑓(1 − ℎ𝑡𝑡𝑞𝑞 − (2 − 𝑓𝑓2)𝑣𝑣𝑛𝑛𝑡𝑡 − 𝑛𝑛𝑡𝑡𝜖𝜖𝑡𝑡𝑒𝑒)𝑤𝑤𝑡𝑡

𝑓𝑓 (11)

where ℎ𝑡𝑡𝑚𝑚𝑤𝑤 = 1 − ℎ𝑡𝑡𝑚𝑚𝑞𝑞 − ℎ𝑡𝑡𝑚𝑚𝑅𝑅 − ℎ𝑡𝑡𝑚𝑚𝑒𝑒 is the time allocated by the male to market production.

In this expression, 𝑒𝑒𝑡𝑡𝑚𝑚ℎ𝑡𝑡𝑚𝑚𝑤𝑤 and 𝑒𝑒𝑡𝑡𝑓𝑓ℎ𝑡𝑡𝑤𝑤 measure labor supply by male and female adults in

efficiency units, and 𝑤𝑤𝑡𝑡𝑚𝑚 and 𝑤𝑤𝑡𝑡

𝑓𝑓 are effective market wages for male and female adults,

respectively.

The household maximizes the utility (1) with respect to 𝑐𝑐𝑡𝑡, 𝑐𝑐𝑡𝑡+1, ℎ𝑡𝑡𝑞𝑞, 𝜖𝜖𝑡𝑡𝑒𝑒, and 𝑛𝑛𝑡𝑡

8 The formulas for children’s human capital accumulation do not include the role of private education spending. However, the mother’s time can be interpreted as comprising private educational spending. The model can be extended to include the allocation of family income to education of children, though the solution of the model becomes much complicated. 9 As in Kim et al (2016), we assume that the savings made by adults who do not survive to old age are confiscated by the government and equally distributed in lump sum to the surviving adults when they become old. Hence the return rate of saving, (1+𝑟𝑟𝑡𝑡)

𝑝𝑝𝐴𝐴 is higher than the actual interest rate, 1 + 𝑟𝑟𝑡𝑡 .

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subject to the constraints (2)-(11). The first order conditions for 𝑐𝑐𝑡𝑡 and 𝑐𝑐𝑡𝑡+1 implies that

(𝑐𝑐𝑡𝑡+1𝑐𝑐𝑡𝑡

)𝜎𝜎 = 1+𝑟𝑟𝑡𝑡+1𝜂𝜂𝑐𝑐(1+𝜌𝜌)

(12)

It is useful to derive the saving rate from (12) as follows:

θ𝑡𝑡 = 1 − 1

1+ 𝑃𝑃𝐴𝐴1+𝑟𝑟𝑡𝑡+1

(1+𝑟𝑟𝑡𝑡+1𝜂𝜂𝑐𝑐(1+𝜌𝜌))1/𝜎𝜎 (13)

Market output is produced by identical firms whose number is normalized to unity.

Each identical firm i’s production function takes the following form:

𝑌𝑌𝑡𝑡𝑖𝑖 = 𝑌𝑌�(𝐸𝐸𝑡𝑡𝑚𝑚𝐻𝐻𝑡𝑡𝑚𝑚𝑤𝑤𝑁𝑁𝑡𝑡𝑚𝑚,𝑖𝑖)𝛼𝛼(𝐸𝐸𝑡𝑡

𝑓𝑓𝐻𝐻𝑡𝑡𝑤𝑤𝑁𝑁𝑡𝑡𝑓𝑓,𝑖𝑖)𝛼𝛼(𝐾𝐾𝑡𝑡𝑖𝑖)1−2𝛼𝛼 (14)

where α ∈ (0,1) is the elasticity of output with respect to male and female effective labor

that is assumed to be the same. Since the representative firm hires labor from the labor

market, it hires male and female workers with average labor productivity (education level)

𝐸𝐸𝑡𝑡𝑚𝑚 and 𝐸𝐸𝑡𝑡𝑓𝑓, respectively. The average male and female adult’s time allocated to market

production is denoted by 𝐻𝐻𝑡𝑡𝑚𝑚𝑤𝑤 and 𝐻𝐻𝑡𝑡𝑤𝑤 and the numbers of male and female workers are

𝑁𝑁𝑡𝑡𝑚𝑚,𝑖𝑖and 𝑁𝑁𝑡𝑡

𝑓𝑓,𝑖𝑖. Finally 𝐾𝐾𝑡𝑡𝑖𝑖 is the amount of capital stock employed by firm i.

Profits of firm i are represented as follows:

Π𝑡𝑡𝑖𝑖 = 𝑌𝑌𝑡𝑡𝑖𝑖 − �𝑤𝑤𝑡𝑡𝑚𝑚𝐸𝐸𝑡𝑡𝑚𝑚𝐻𝐻𝑡𝑡𝑚𝑚𝑁𝑁𝑡𝑡

𝑚𝑚,𝑖𝑖 + 𝑤𝑤𝑡𝑡𝑓𝑓𝐸𝐸𝑡𝑡

𝑓𝑓𝐻𝐻𝑡𝑡𝑤𝑤𝑁𝑁𝑡𝑡𝑓𝑓,𝑖𝑖� − 𝑟𝑟𝑡𝑡𝐾𝐾𝑡𝑡𝑖𝑖 (15)

where the price of the marketed good is normalized to unity and 𝑟𝑟𝑡𝑡 is the rental rate of

capital that is identical to the rate of return to savings. The firm, taking input prices as given,

maximizes profits with respect to the number of male and female workers and capital.

As in Kim et al. (2016), we assume that there is discrimination in the labor market

against female workers: while male workers receive their marginal product, female workers

receive a faction d ∈ (0,1) of their marginal product. For simplicity, we assume that firms

do not distribute to households the profits accrued due to female discrimination in the labor

11

market. Then the optimal choices of the firm for labor and capital satisfy the following

equations:

𝑤𝑤𝑡𝑡𝑚𝑚 = 𝛼𝛼𝑌𝑌𝑡𝑡

𝑖𝑖

𝐸𝐸𝑡𝑡𝑚𝑚𝐻𝐻𝑡𝑡

𝑚𝑚𝑚𝑚𝑁𝑁𝑡𝑡𝑚𝑚,𝑖𝑖, 𝑤𝑤𝑡𝑡

𝑓𝑓 = 𝑑𝑑𝛼𝛼𝑌𝑌𝑡𝑡𝑖𝑖

𝐸𝐸𝑡𝑡𝑓𝑓𝐻𝐻𝑡𝑡𝑚𝑚𝑁𝑁𝑡𝑡

𝑓𝑓,𝑖𝑖, 𝑟𝑟𝑡𝑡 = (1 − 2𝛼𝛼) 𝑌𝑌𝑡𝑡𝑖𝑖

𝐾𝐾𝑡𝑡𝑖𝑖 (16)

In equilibrium, 𝑁𝑁𝑡𝑡𝑚𝑚,𝑖𝑖 = 𝑁𝑁𝑡𝑡𝑚𝑚 , 𝑁𝑁𝑡𝑡

𝑓𝑓,𝑖𝑖 = 𝑁𝑁𝑡𝑡𝑓𝑓and 𝐾𝐾𝑡𝑡𝑖𝑖 = 𝐾𝐾𝑡𝑡 for all i and the aggregate output is,

𝑌𝑌𝑡𝑡 = ∫ 𝑌𝑌𝑡𝑡𝑖𝑖10 = 𝑌𝑌�(𝐸𝐸𝑡𝑡𝑚𝑚𝐻𝐻𝑡𝑡𝑚𝑚𝑤𝑤𝑁𝑁𝑡𝑡𝑚𝑚)𝛼𝛼(𝐸𝐸𝑡𝑡

𝑓𝑓𝐻𝐻𝑡𝑡𝑤𝑤𝑁𝑁𝑡𝑡𝑓𝑓)𝛼𝛼(𝐾𝐾𝑡𝑡)1−2𝛼𝛼 (17)

From (14) and the equilibrium conditions, the following relation holds between 𝑤𝑤𝑡𝑡𝑚𝑚 and 𝑤𝑤𝑡𝑡

𝑓𝑓:

𝑤𝑤𝑡𝑡𝑚𝑚𝐸𝐸𝑡𝑡𝑚𝑚𝑤𝑤𝐻𝐻𝑡𝑡𝑚𝑚 = 𝑑𝑑−1𝑤𝑤𝑡𝑡

𝑓𝑓𝐸𝐸𝑡𝑡𝑓𝑓𝐻𝐻𝑡𝑡𝑤𝑤 (18)

In equilibrium the following equations hold: 𝑒𝑒𝑡𝑡𝑚𝑚 = 𝐸𝐸𝑡𝑡𝑚𝑚 , 𝑒𝑒𝑡𝑡𝑓𝑓 = 𝐸𝐸𝑡𝑡

𝑓𝑓 , ℎ𝑡𝑡𝑚𝑚𝑤𝑤 = 𝐻𝐻𝑡𝑡𝑚𝑚𝑤𝑤 and

ℎ𝑡𝑡𝑤𝑤 = 𝐻𝐻𝑡𝑡𝑤𝑤.

The government finances its expenditure on education, 𝐺𝐺𝑡𝑡 and on unproductive

usage, 𝑈𝑈𝑡𝑡 by taxing the wage income. 10 We assume that the expenditure on the

unproductive usage is proportional to that on education: 𝑈𝑈𝑡𝑡 = ∅𝐺𝐺𝑡𝑡. Further we assume that

the government budget is balanced every period:

𝐺𝐺𝑡𝑡 + 𝑈𝑈𝑡𝑡 = 𝜏𝜏�𝐸𝐸𝑡𝑡𝑚𝑚𝐻𝐻𝑡𝑡𝑚𝑚𝑁𝑁𝑡𝑡𝑚𝑚𝑤𝑤𝑡𝑡𝑚𝑚 + 𝐸𝐸𝑡𝑡

𝑓𝑓𝐻𝐻𝑡𝑡𝑤𝑤𝑁𝑁𝑡𝑡𝑓𝑓𝑤𝑤𝑡𝑡

𝑓𝑓� (19)

where 𝜏𝜏 is the tax rate of government expenditure. Then

(1 + ∅)𝐺𝐺𝑡𝑡 = 𝜏𝜏�𝐸𝐸𝑡𝑡𝑚𝑚𝐻𝐻𝑡𝑡𝑚𝑚𝑁𝑁𝑡𝑡𝑚𝑚𝑤𝑤𝑡𝑡𝑚𝑚 + 𝐸𝐸𝑡𝑡

𝑓𝑓𝐻𝐻𝑡𝑡𝑤𝑤𝑁𝑁𝑡𝑡𝑓𝑓𝑤𝑤𝑡𝑡

𝑓𝑓� (20)

or

(1 + ∅)𝑔𝑔𝑡𝑡 = 𝜏𝜏�𝐸𝐸𝑡𝑡𝑚𝑚𝐻𝐻𝑡𝑡𝑚𝑚𝑤𝑤𝑡𝑡𝑚𝑚 + 𝐸𝐸𝑡𝑡

𝑓𝑓𝐻𝐻𝑡𝑡𝑤𝑤𝑤𝑤𝑡𝑡𝑓𝑓� (21)

where 𝑔𝑔𝑡𝑡 ≡𝐺𝐺𝑡𝑡𝑁𝑁𝑡𝑡𝑓𝑓 = 𝐺𝐺𝑡𝑡

𝑁𝑁𝑡𝑡/2.

10 The model can be easily extended to allow nondistortionary revenue financing public education expenditures or unproductive government spending reallocated to the education sector. This extension will produce a more positive contribution to economic growth in the form of an increase in government education spending.

12

In equilibrium, from (16) and (18)

(1 + ∅)𝑔𝑔𝑡𝑡 = 𝜏𝜏�𝐸𝐸𝑡𝑡𝑚𝑚𝐻𝐻𝑡𝑡𝑚𝑚𝑤𝑤𝑡𝑡𝑚𝑚 + 𝐸𝐸𝑡𝑡

𝑓𝑓𝐻𝐻𝑡𝑡𝑤𝑤𝑤𝑤𝑡𝑡𝑓𝑓� = 𝜏𝜏�𝑒𝑒𝑡𝑡𝑚𝑚ℎ𝑡𝑡𝑚𝑚𝑤𝑤𝑡𝑡

𝑚𝑚 + 𝑒𝑒𝑡𝑡𝑓𝑓ℎ𝑡𝑡𝑤𝑤𝑤𝑤𝑡𝑡

𝑓𝑓�=τ(1+𝑑𝑑−1)𝑒𝑒𝑡𝑡𝑓𝑓ℎ𝑡𝑡𝑤𝑤𝑤𝑤𝑡𝑡

𝑓𝑓

= 2𝜏𝜏(1 + 𝑑𝑑) 𝛼𝛼 𝑌𝑌𝑡𝑡𝑁𝑁𝑡𝑡

, or (22)

(1 − ∅)𝐺𝐺𝑡𝑡 = 𝜏𝜏(1 + 𝑑𝑑) 𝛼𝛼𝑌𝑌𝑡𝑡 (23)

A competitive equilibrium satisfies the following three conditions:

(i) The household maximizes utility (1) with respect to 𝑐𝑐𝑡𝑡, 𝑐𝑐𝑡𝑡+1,𝑛𝑛𝑡𝑡, ℎ𝑡𝑡𝑤𝑤,ℎ𝑡𝑡𝑞𝑞 ,ℎ𝑡𝑡𝑅𝑅 and ℎ𝑡𝑡𝑒𝑒.

(ii) The firm maximizes profits with respect to 𝑁𝑁𝑡𝑡𝑚𝑚,𝑖𝑖,𝑁𝑁𝑡𝑡

𝑓𝑓,𝑖𝑖 and 𝐾𝐾𝑡𝑡𝑖𝑖.

(iii) Markets cleared. In particular the asset-market clearing condition requires that total

savings by all households (0.5𝑁𝑁𝑡𝑡) in period t are equal to total capital stock at the beginning

of period (t+1): 0.5𝑁𝑁𝑡𝑡𝑠𝑠𝑡𝑡 = 𝑁𝑁𝑡𝑡𝑓𝑓𝑠𝑠𝑡𝑡 = 𝐾𝐾𝑡𝑡+1.

In the balanced growth path, it can be easily verifiable that 𝑌𝑌𝑡𝑡𝑁𝑁𝑡𝑡

and 𝐾𝐾𝑡𝑡𝑁𝑁𝑡𝑡

grow at the

same rate as 𝑒𝑒𝑡𝑡𝑓𝑓 . Hence the female education (that is the same as the male education) is the

key to perpetual growth.

The growth rate of per capita GDP in steady state is11:

1 + 𝛾𝛾𝑌𝑌/𝑁𝑁 = 2𝑌𝑌�(1 − 𝑓𝑓ℎ𝑞𝑞∗ − 𝑓𝑓2𝑣𝑣𝑛𝑛∗ − 𝑓𝑓3𝑛𝑛∗𝜖𝜖𝑒𝑒∗)𝛼𝛼 (1 − ℎ𝑞𝑞∗ − (2 − 𝑓𝑓2)𝑣𝑣𝑛𝑛∗ −

𝑛𝑛∗𝜖𝜖𝑒𝑒∗)𝛼𝛼(𝑘𝑘∗)−2𝛼𝛼𝑑𝑑𝛼𝛼Φ𝜃𝜃∗(𝑛𝑛∗)−1 (24)

where the variables with * are steady state values and 𝑘𝑘𝑓𝑓∗ = ( 𝐾𝐾𝑒𝑒𝑓𝑓𝑁𝑁𝑓𝑓

)∗.

2.2 Calibration and Balanced Growth Path

Most parameter values are from the macroeconomics literature and Kim et al. (2016).

Some of our parameters are derived from the calibration of our model to fit into its steady-

state values, which are derived from the average values from Korea for period 2005-2010 as

11 See Appendix B for the derivation.

13

reported in the World Development Indicators by the World Bank, Bank of Korea data, and

Korea Time Use Survey (2009) data. The values are as follows:

(1) Fertility: 1.17

(2) Annual per capita income growth rate: 3.6%

(3) Net private saving rate (% of disposable income): 16.10%12

(4) Female and male labor force participation rate: 54.43% and 75.92%

(5) Wife-husband ratio of child rearing time: 5.113 (51 min. a day by wife and 10 min. by

husband)

(6) Wife-husband ratio of child education time: 3.25 (26 min. a day by wife and 8 min. by

husband).

f2 is derived from equation (2 − f2) /f2 = 5.1 while f3 is derived from equation 1/f3 = 3.25.

Since the male labor force participation rate in our model is (1 − 𝑓𝑓1ℎ𝑡𝑡𝑞𝑞 − 𝑓𝑓2𝑣𝑣𝑛𝑛𝑡𝑡 − 𝑓𝑓3𝑛𝑛𝑡𝑡𝜖𝜖𝑡𝑡𝑒𝑒),

parameter f1 can be estimated from the equation: 𝑓𝑓1ℎ𝑡𝑡𝑞𝑞 = 1 − 0.7592 −𝑓𝑓2𝑣𝑣𝑛𝑛𝑡𝑡 − 𝑓𝑓3𝑛𝑛𝑡𝑡𝜖𝜖𝑡𝑡𝑒𝑒,

where ℎ𝑡𝑡𝑞𝑞, 𝑛𝑛𝑡𝑡, and 𝜖𝜖𝑡𝑡𝑒𝑒 are endogenously determined in our model. From the calibration with

other average values, we are able to pin down the following parameter values: f1 = 0.5897, f2

= 0.3279, f3 = 0.3077, v = 2.8099, ρ = 0.5982, �̅�𝑒 = 4.2797, and 𝑞𝑞� = 23.6313.14 Table 3

reports the parameter values used for the calibration, and Table 4 presents the steady-state

values of key variables in the model economy.

3. Estimation of the Economic Effects of Gender Inequality in Korea

12 Net saving rate is private saving rate (22.10%) minus depreciation (6%). 13 Childrearing time includes time spent on washing, feeding, sending off to school, putting in bed, and transportation of children. Child education time includes time spent on helping homework, teaching, and reading. 14 The estimated ρ should be interpreted as the discount rate which comprises the relative preference parameter for consumption ηc. as well as time discount rate over 30 years.

14

3.1 Output Costs of Gender Inequality

We can measure the output costs of gender inequality by comparing the

performances of the benchmark case with those of a hypothetical Korean economy with no

gender inequality. In the hypothetical gender-equality case, males and females have the same

opportunities and power at home, in education, and in the labor markets.

Table 5 reports the steady-state values of various key variablesfertility, female

labor force participation, and per capita output growth ratein the benchmark case (shown in

model (i)) as well as in several hypothetical cases.

Model (ii) of this table illustrates the alternative steady-state of the economy with

complete gender equality (𝑑𝑑 = 1, and 𝑓𝑓1 = 𝑓𝑓2 = 𝑓𝑓3 = 1). According to the simulation results,

the female labor market participation rate increases from 54.4 to 67.5 with complete gender

equality. Note that in our framework, the labor force participation rates for males and females

are equal with no gender bias at home and labor market. Per capita output growth rate in the

new steady-state increases to a higher value. The results show that by eliminating the gender

inequality, the annual growth rates of per capita income can be enhanced by approximately

0.5% point, by increasing from the current level of 3.6%. to 4.1%. We also find that the

fertility rate becomes 0.98, lower than the value in the benchmark case, 1.17.

The table also presents the new steady state values that would be reached by the

Korean economy if one of the four inequalities or the inequalities both in home production

and labor market are eliminated for a comparison with the case of complete gender equality.

The result in model (iii), for example, shows that with the complete elimination of the gender

discrimination in labor market alone (i.e., d=1), the female labor market participation rate

increases from 54.4% to 59.3%, and per capita income growth increases from 3.6% to 4.3%

on average over a generation.

15

Interestingly, removing only the gender inequality in home production (𝑓𝑓1 = 1) or

education (𝑓𝑓3 = 1) lowers the growth rate of per capita income. The decrease in per capita

output growth rate is mainly due to a decrease in male labor supply. As husbands increase the

time they allocate to home production, child rearing, and education with perfect gender

equality in home production or in education, they will spend less time in the labor market.

Another reason for the lower income growth rate per person in the case of perfect gender

equality in home production (𝑓𝑓1 = 1) is an increase in fertility and thus also in the

population growth rate.

3.2 Gender-based Policies

We consider the following three policies to promote gender equality:

(i) Lower discrimination in the labor market (that is, raising 𝑑𝑑),

(ii) Increase the time spent by a male on child rearing (raising f2),

(iii) Lower time cost for child rearing (lowering 𝑣𝑣).

In Figure 2, we illustrate changes in three key variables of the most interest—fertility

rate, female labor market participation rate, and per capita income growth—when the three

policies are implemented.

Lowering the discrimination in the labor market by changing the value of d from 0.6

to 0.7 increases the growth rate of per capita output by about 0.2% point. When the distortion

in the labor market is reduced, the female’s time allocated to market production significantly

increases, contributing to the increase in per capita output growth. In this case, the fertility is

lowered as females allocate more time to market production.

If males increase time for child rearing, i.e., raising f2 (from 0.328 to 0.667), both

female labor market participation and growth rate of per capita output increase. In this case,

the fertility rate decreases.

16

In contrast, when the rearing time needed per child v is lowered from 2.810 to 2.5,

the growth rate of per capita output decreases. Since a decrease in v implies that the cost

involved with increasing the quantity of children is lowered, the optimal decision is to

increase the fertility. In this case, the increase in the fertility rate dominates the increase in

aggregate output, eventually lowering the growth rate of the per capita output.

4. Concluding Remarks

This paper provides a theoretical framework that can explain the determination of

female labor market participation, human capital accumulation, and economic growth in the

Korean economy. We employ this framework to quantitatively analyze the output cost of

gender inequality. Our results indicate that the output cost of gender inequality is quite

sizable. If the gender inequality is completely eliminated, the female labor force participation

rate increases from 54.4% to 67.5%, and the annual per capita income growth rises from 3.6%

to 4.1% on average over a generation. The increase in the economic growth rate implies that

with the complete elimination of gender inequality, per capita income will become

approximately 15% higher over one generation. We believe that this growth enhancing effect

of gender equality is comparable to that of other types of policies contemplated in the Korea

economy, such as increasing public infrastructure investment and removing unnecessary

regulations.

Among various policy measures related with gender equality that we contemplated in

our study, we find that the most effective policy in terms of enhancing the growth rate of per

capita income is eliminating the discrimination in the labor market. Policies that attempt to

mitigate gender inequalities by reducing women’s time allocated to home production, child

rearing, and education would be helpful for enhancing growth when they are combined with

17

the reduction of fundamental discrimination in the labor market and are designed to minimize

negative influences on the male’s labor market participation.

One intriguing lesson from our analysis for policy-making is that a particular policy

to meet a specific goal can sometimes produce an adverse effect in terms of another goal of

our concerns. One example of this point is the policy to eliminate discrimination against

women in the labor market. On one hand, this policy will encourage more women to

participate in the labor market and increase female labor force participation. However, it will

also raise the opportunity cost of time for women, which will likely lower fertility and thus

exacerbate the aging population problem. We can also show that childcare subsidies will

lower the cost of childrearing and therefore increase fertility while this policy will lower

female labor market participation. Similarly, a public policy that promotes males’ engagement

in child rearing can increase female labor market participation and per capita income growth,

but lower fertility.

Since the early 2000s, the relationship between female labor market participation and

fertility among OECD countries (based on the cross-sectional data) has been changed from a

negative one to a positive one. The OECD data also show a positive relationship between the

gender equality index and fertility in recent years. The results from our analysis indicate that

these recent changes are not due to one particular policy but rather due to a combination of

policies that are well coordinated. It would be wise for the Korean government to implement

multiple public policies together in order to achieve a multiple number of goals.

Our analysis has left a number of important issues related to gender inequality

unaddressed. For example, our model does not consider the glass ceiling in promotion for

women in business and in the public sector, social norms against gender equality, endogenous

determination of bargaining power between wives and husbands, and the like.

18

While it is desirable to model gender inequality that arises endogenously from more

fundamental structures of the economy, we treat gender inequality as exogenous in our model

because our main goal is to quantify the impact of gender equality policies on female labor

force participation and economic growth. We leave this issues and others to future work.

19

REFERENCES

Agénor, P. R. (2016). “A computable OLG model for gender and growth policy analysis.” , Macroeconomic Dynamics, forthcoming in June..

Agenor, P.-R., and O. Canuto, (2015) "Gender Equality and Economic Growth in Brazil: A Long-Run Analysis," Journal of Macroeconomics, 43, March, 155-72. Barro, R. J., and Lee, J. W. (2013). “A new data set of educational attainment in the world, 1950–2010.” Journal of Development Economics, 104, 184-198.

Becker, G. S., Murphy, K. M., & Tamura, R. (1990). “Human capital fertility and economic growth.” Journal of Political Economy, 98(5 Pt. 2), S12-37.

Bekhouch, Y., Hausmann, R., Tyson, L. D., & Zahidi, S. (2013). The Global Gender Gap Report 2013.

Cho, Sun-Joo. (2009) “Earned Income Tax Credit and Female Labor Supply.” Quarterly Journal of Labor Policy, 9(3), 29~54, in Korean.

Choi, Seng-Eun. (2011) “Child Care Subsidies and the Labor Supply of Married Women.” Korean Social Security Studies, 27(2), 85~105, in Korean.

Ehrlich, Isaac and Lui, Francis T. (1991). “Intergenerational Trade, Longevity, and Economic Growth.” Journal of Political Economy, 99, 1029-1059.

Galor, O., and Weil, D. N. (1996). “The gender gap, fertility, and growth.” American Economic Review, 85(3), 374–387.

GMI Ratings. “GMI Ratings’ 2013 Women On Board Survey”, April 2013.

Heo, Su-Yeon. (2008) “Research on Time Use for Housework of Women and Men in Double Income Households,” Journal of Korean Women's Studies, 24(3), 177-201, in Korean.

Kim, Hyunsook, and Sung, Myung Jae. (2007) “The Effects of Introduction of Child Tax Credit on the Married Women's Labor Supply in Korea,” Korean Journal of Public Economics, 12(1), 75-117, in Korean.

Kim, J. “Income and Wage Effects on Labor Force Participation: Evidence from Married Women in Korea,” mimeo, Korea University, 2015.

Kim J., J.-W. Lee and K. Shin. (2016) “A Model of Gender Inequality and Economic Growth,” ADB Economics Working Paper Series No. 475.

Kim, Ji Kyung, and Cho, You Hyun. (2003) “Analysis of Married Women’s Return to the Workforce Following First Childbirth.” Korean Journal of Labor Economics, 26(3), 181-207, in Korean.

Kim, Jungho. (2012) “Parental Leave and Female Labor Supply in Korea,” KDI Journal of Economic Policy, 2012(1), in Korean.

Lagerlof, N.P. (2003). “Gender equality and long run growth.” Journal of Economic Growth,

20

8(4), 403-426.

Ministry of Education, Science and Technology. Statistical Yearbook of Education, various years.

Ministry of Employment and Labor. Survey on Labor Conditions by Type of Employment, 2010.

OECD, statistical data available at http://stats.oecd.org.

Statistics Korea, Annual Report on the Economically Active Population Survey, 2011.

Statistics Korea, Social Indicators in Korea, various years.

Statistics Korea, Korea Time Use Survey, 2009.

Woo, Seokjin. (2008) “The outcomes of foster care in South Korea ten years after its foundation: A comparison with institutional care,” Journal of Korean Economic Analysis, 14(3), 55-105, in Korean.

21

APPENDIX A

The Korean government has been pursuing various policies to improve the welfare of women in Korea and to reduce gender inequality. A parallel objective of these policies is to encourage childbearing owing to the accelerated aging of Korean society that has been accompanied by extremely low fertility rates during the last few decades.

With these policies, the Korean government in general aims to provide women with a better environment for childbearing and child rearing so as to encourage and enable them to return to the labor market even as they are raising children.

The policies can be categorized into three types:

1. The first type of policies includes those aimed at helping women give birth successfully and at lowering the cost of childbearing for couples. This type of policies include: (a) maternity and paternity leave (for childbirth), (b) childcare leave of absence (for children ages 0-6) for a maximum of 1 year with pay, and (c) reduced work hours during child rearing (for children ages 0-6) for a maximum of 1 year combined with child care leave of absence.

2. The second type of policies aims at providing parents with more reliable childcare facilities for children in grade schools. These are: (a) incentives for firms to provide childcare centers at work: implemented in June 2013, (b) providing public childcare centers, (c) financial support for childcare, and (d) encouraging private and public childcare centers with financial incentives to offer flexible hours for childcare (starting July 2014).

3. The third type of policies aims at giving companies incentives to hire back women after childbirth or child care. At present, private firms can receive tax subsidies if they employ female workers who are trying to return to the labor market after childrearing.

These policies have been successful to some extent in promoting female labor market participation. For instance, in 2014, the female employment rate among women from age 15 and over rose above the 50% mark—50.4% to be exact—for the first time (Economically Active Population Survey, 2014). However, Korea’s female employment rate is still significantly lower than that of many advanced economies like Canada (69.2%), Japan (60.7%), and Sweden (71.8%). This leaves a lot of room for the Korean government to implement better policies for encouraging women’s participation in the labor market.

APPENDIX B

In this appendix, we derive equations needed to solve the steady states. Then we calculate the balanced growth rate.

The household problem is to maximize the household utility function:

𝑈𝑈𝑡𝑡 = 𝜂𝜂𝑐𝑐1

1 − 𝜎𝜎𝑐𝑐𝑡𝑡1−𝜎𝜎 + 𝜂𝜂𝑞𝑞

11 − 𝜎𝜎

𝑞𝑞𝑡𝑡1−𝜎𝜎

22

+𝜂𝜂𝑒𝑒 �1

1−𝜎𝜎��𝑛𝑛𝑡𝑡

2�𝛿𝛿𝑒𝑒𝑡𝑡+1𝑚𝑚 �

1−𝜎𝜎+ 1

1−𝜎𝜎��𝑛𝑛𝑡𝑡

2�𝛿𝛿𝑒𝑒𝑡𝑡+1𝑓𝑓 �

1−𝜎𝜎� + 𝑝𝑝𝐴𝐴

1+𝜌𝜌1

1−𝜎𝜎𝑐𝑐𝑡𝑡+11−𝜎𝜎 (A1)

Subject to

𝑞𝑞𝑡𝑡 = 𝑞𝑞�(1 + 𝑓𝑓1)𝛾𝛾(ℎ𝑡𝑡𝑞𝑞)𝛾𝛾𝑒𝑒𝑡𝑡

𝑓𝑓 (A2)

(1 − 𝜏𝜏)𝑒𝑒𝑡𝑡𝑚𝑚�1 − 𝑓𝑓1ℎ𝑡𝑡𝑞𝑞 − 𝑓𝑓2𝑣𝑣𝑛𝑛𝑡𝑡 − 𝑓𝑓3𝑛𝑛𝑡𝑡𝜖𝜖𝑡𝑡𝑒𝑒�𝑤𝑤𝑡𝑡𝑚𝑚

+(1 − 𝜏𝜏)𝑒𝑒𝑡𝑡𝑓𝑓�1 − ℎ𝑡𝑡

𝑞𝑞 − (2 − 𝑓𝑓2)𝑣𝑣𝑛𝑛𝑡𝑡 − 𝑛𝑛𝑡𝑡𝜖𝜖𝑡𝑡𝑒𝑒�𝑤𝑤𝑡𝑡𝑓𝑓 − 𝑐𝑐𝑡𝑡 −

𝑝𝑝𝐴𝐴𝑐𝑐𝑡𝑡+11+𝑟𝑟𝑡𝑡+1

= 0 (A3)

𝑒𝑒𝑡𝑡+1𝑚𝑚 = �̅�𝑒( 𝜇𝜇𝐺𝐺𝑡𝑡𝑛𝑛𝑡𝑡𝑎𝑎𝑁𝑁𝑡𝑡/2

)𝜈𝜈1[𝑒𝑒𝑡𝑡𝑓𝑓]1−𝜈𝜈1((1 + 𝑓𝑓3)𝜖𝜖𝑡𝑡𝑒𝑒)𝜈𝜈2 (A4)

𝑒𝑒𝑡𝑡+1𝑓𝑓 = �̅�𝑒( 𝜇𝜇𝐺𝐺𝑡𝑡

𝑛𝑛𝑡𝑡𝑎𝑎𝑁𝑁𝑡𝑡/2

)𝜈𝜈1[𝑒𝑒𝑡𝑡𝑓𝑓]1−𝜈𝜈1((1 + 𝑓𝑓3)𝜖𝜖𝑡𝑡𝑒𝑒)𝜈𝜈2 (A5)

The first-order optimality conditions derived from the household problem are:

(𝑐𝑐𝑡𝑡) 𝜂𝜂𝑐𝑐𝑐𝑐𝑡𝑡−𝜎𝜎 = 𝜆𝜆

(𝑐𝑐𝑡𝑡+1) 𝑝𝑝𝐴𝐴1+𝜌𝜌

𝑐𝑐𝑡𝑡+1−𝜎𝜎 = 𝜆𝜆 𝑝𝑝𝐴𝐴1+𝑟𝑟𝑡𝑡+1

, or (𝑐𝑐𝑡𝑡+1𝑐𝑐𝑡𝑡

)𝜎𝜎 = 1+𝑟𝑟𝑡𝑡+1𝜂𝜂𝑐𝑐(1+𝜌𝜌)

(A6)

(ℎ𝑡𝑡𝑞𝑞) 𝜂𝜂𝑞𝑞𝑞𝑞�1−𝜎𝜎𝛾𝛾(1 + 𝑓𝑓)𝛾𝛾(1−𝜎𝜎)(ℎ𝑡𝑡

𝑞𝑞)(1−𝜎𝜎)𝛾𝛾−1�𝑒𝑒𝑡𝑡𝑓𝑓�

1−𝜎𝜎= 𝜂𝜂𝑐𝑐𝑐𝑐𝑡𝑡−𝜎𝜎(1 − 𝜏𝜏)(𝑓𝑓1𝑒𝑒𝑡𝑡𝑚𝑚𝑤𝑤𝑡𝑡

𝑚𝑚 + 𝑒𝑒𝑡𝑡𝑓𝑓𝑤𝑤𝑡𝑡

𝑓𝑓) (A7)

(𝜖𝜖𝑡𝑡𝑒𝑒) 𝜂𝜂𝑒𝑒 ��nt2�δ

et+1m �

-σ�nt

2�δ

e� � μGtntaNt2

�ν1

(etf)

1-ν1(1 + 𝑓𝑓3)𝜈𝜈2ν2(ϵte)ν2-1

+ 𝜂𝜂𝑒𝑒 ��nt2�δ

et+1f �−σ

�nt2�δ

e� �µGt

ntaNt2

�

ν1

(etf)1−ν1(1 + 𝑓𝑓3)𝜈𝜈2ν2(ϵte)ν2−1

= ηcct−σ(1 − 𝜏𝜏)[𝑒𝑒𝑡𝑡𝑚𝑚(𝑓𝑓3𝑛𝑛𝑡𝑡)𝑤𝑤𝑡𝑡𝑚𝑚+𝑒𝑒𝑡𝑡

𝑓𝑓(𝑛𝑛𝑡𝑡𝜖𝜖𝑡𝑡𝑒𝑒)𝑤𝑤𝑡𝑡𝑓𝑓] (A8)

(𝑛𝑛𝑡𝑡) 𝜂𝜂𝑒𝑒(et+1m )1−σ �12�δ(1−σ)

δ(nt)δ(1−σ)−1 + 𝜂𝜂𝑒𝑒�et+1f �1−σ

�12�δ(1−σ)

δ(nt)δ(1−σ)−1

= ηcct−σ(1− τ)[etfwtf(ϵte + (2 − 𝑓𝑓2)v) + 𝑒𝑒𝑡𝑡𝑚𝑚wt

m(𝑓𝑓3ϵte + 𝑓𝑓2v)] . (A9)

Since 𝑒𝑒𝑡𝑡𝑚𝑚 = 𝐸𝐸𝑡𝑡𝑚𝑚 , 𝑒𝑒𝑡𝑡𝑓𝑓 = 𝐸𝐸𝑡𝑡

𝑓𝑓 , ℎ𝑡𝑡𝑚𝑚 = 𝐻𝐻𝑡𝑡𝑚𝑚 , ℎ𝑡𝑡𝑓𝑓 = 𝐻𝐻𝑡𝑡

𝑓𝑓 , 𝑌𝑌𝑡𝑡𝑖𝑖 = 𝑌𝑌𝑡𝑡 and 𝑁𝑁𝑡𝑡𝑚𝑚,𝑖𝑖 = 𝑁𝑁𝑡𝑡

𝑓𝑓,𝑖𝑖 =12𝑁𝑁𝑡𝑡 in an equilibrium, we have

𝑤𝑤𝑡𝑡𝑓𝑓 = 2𝑑𝑑𝛼𝛼

𝑒𝑒𝑡𝑡𝑓𝑓ℎ𝑡𝑡𝑚𝑚

𝑌𝑌𝑡𝑡𝑁𝑁𝑡𝑡

, and (A10)

𝑤𝑤𝑡𝑡𝑚𝑚 = 2𝛼𝛼𝑌𝑌𝑡𝑡

𝑒𝑒𝑡𝑡𝑚𝑚ℎ𝑡𝑡

𝑚𝑚𝑚𝑚𝑁𝑁𝑡𝑡 = 2𝛼𝛼

𝑒𝑒𝑡𝑡𝑓𝑓(1−𝑓𝑓1ℎ𝑡𝑡

𝑞𝑞−𝑓𝑓2𝑣𝑣𝑛𝑛𝑡𝑡−𝑓𝑓3𝑛𝑛𝑡𝑡𝜖𝜖𝑡𝑡𝑒𝑒)

𝑌𝑌𝑡𝑡𝑁𝑁𝑡𝑡

. (A11)

23

(i) Dynamics for the population size 𝑵𝑵𝒕𝒕

The number of adults next period 𝑁𝑁𝑡𝑡+1 is the surviving children born at time t. Since the number of households at time t is 𝑁𝑁𝑡𝑡

2 and each household gives birth to 𝑛𝑛𝑡𝑡 that will survive

with probability 𝑝𝑝𝑐𝑐, the dynamics of 𝑁𝑁𝑡𝑡 follows:

𝑁𝑁𝑡𝑡+1 = 𝑛𝑛𝑡𝑡𝑁𝑁𝑡𝑡2

(A12)

(ii) Savings

For deriving the saving rate, we can use equations (7) and (8).

𝑐𝑐𝑡𝑡 + 𝑝𝑝𝐴𝐴𝑐𝑐𝑡𝑡+1(1+𝑟𝑟𝑡𝑡+1) = (1 − 𝜏𝜏)𝑤𝑤𝑡𝑡

𝐻𝐻 (A13)

Substituting (A6) into (A13) yields,

𝑐𝑐𝑡𝑡 + 𝑝𝑝𝐴𝐴1+𝑟𝑟𝑡𝑡+1

( 1+𝑟𝑟𝑡𝑡+1𝜂𝜂𝑐𝑐(1+𝜌𝜌)

)1/𝜎𝜎𝑐𝑐𝑡𝑡 = (1 − 𝜏𝜏)𝑤𝑤𝑡𝑡𝐻𝐻, and therefore (A14)

𝑐𝑐𝑡𝑡 = 11+ 𝑝𝑝𝐴𝐴

1+𝑟𝑟𝑡𝑡+1(1+𝑟𝑟𝑡𝑡+1𝜂𝜂𝑐𝑐(1+𝜌𝜌))1/𝜎𝜎 (1 − 𝜏𝜏)𝑤𝑤𝑡𝑡

𝐻𝐻 (A15)

Hence the saving rate 𝜃𝜃𝑡𝑡 is

𝜃𝜃𝑡𝑡 = 1 − 11+ 𝑝𝑝𝐴𝐴

1+𝑟𝑟𝑡𝑡+1(1+𝑟𝑟𝑡𝑡+1𝜂𝜂𝑐𝑐(1+𝜌𝜌))1/𝜎𝜎 (A16)

Since 𝑒𝑒𝑡𝑡𝑚𝑚 = 𝐸𝐸𝑡𝑡𝑚𝑚 , 𝑒𝑒𝑡𝑡𝑓𝑓 = 𝐸𝐸𝑡𝑡

𝑓𝑓 , ℎ𝑡𝑡𝑚𝑚 = 𝐻𝐻𝑡𝑡𝑚𝑚 and ℎ𝑡𝑡𝑓𝑓 = 𝐻𝐻𝑡𝑡

𝑓𝑓 is held in equilibrium, total gross wage income for the household becomes

𝑤𝑤𝑡𝑡𝑚𝑚𝑒𝑒𝑡𝑡𝑚𝑚ℎ𝑡𝑡𝑚𝑚𝑤𝑤 = 𝑑𝑑−1𝑤𝑤𝑡𝑡

𝑓𝑓𝑒𝑒𝑡𝑡𝑓𝑓ℎ𝑡𝑡𝑤𝑤 (A17)

Then the budget constraint for the household becomes

𝑤𝑤𝑡𝑡𝐻𝐻 = 𝑒𝑒𝑡𝑡𝑚𝑚ℎ𝑡𝑡𝑚𝑚𝑤𝑤𝑤𝑤𝑡𝑡

𝑚𝑚 + 𝑒𝑒𝑡𝑡𝑓𝑓ℎ𝑡𝑡𝑤𝑤𝑤𝑤𝑡𝑡

𝑓𝑓 = (1+𝑑𝑑−1)𝑒𝑒𝑡𝑡𝑓𝑓ℎ𝑡𝑡𝑤𝑤𝑤𝑤𝑡𝑡

𝑓𝑓 (A18)

Total savings 𝑆𝑆𝑡𝑡 in equilibrium are therefore

𝑆𝑆𝑡𝑡 = 𝜃𝜃𝑡𝑡(1 − 𝜏𝜏)(1+𝑑𝑑−1)𝑒𝑒𝑡𝑡𝑓𝑓ℎ𝑡𝑡𝑤𝑤𝑤𝑤𝑡𝑡

𝑓𝑓 = 𝜃𝜃𝑡𝑡 Φ𝑒𝑒𝑡𝑡𝑓𝑓�1 − ℎ𝑡𝑡

𝑞𝑞 − (2 − 𝑓𝑓2)𝑣𝑣𝑛𝑛𝑡𝑡 − 𝑛𝑛𝑡𝑡𝜖𝜖𝑡𝑡𝑒𝑒�𝑤𝑤𝑡𝑡𝑓𝑓 (A19)

where Φ = (1 − 𝜏𝜏)(1 + 𝑑𝑑−1). We can also show that the interest rate is

𝑟𝑟𝑡𝑡+1 = (1 − 2𝛼𝛼) 𝑌𝑌𝑡𝑡+1𝐾𝐾𝑡𝑡+1

(A20)

(iii) Dynamics for the capital stock 𝑲𝑲𝒕𝒕

From the condition for the equilibrium in section 3.1, we have

𝐾𝐾𝑡𝑡+1 = 0.5�𝑁𝑁𝑡𝑡𝑚𝑚 + 𝑁𝑁𝑡𝑡𝑓𝑓�𝑆𝑆𝑡𝑡 = 𝑁𝑁𝑡𝑡

𝑓𝑓𝑆𝑆𝑡𝑡 = Φ𝑁𝑁𝑡𝑡𝑓𝑓𝜃𝜃𝑡𝑡𝑒𝑒𝑡𝑡

𝑓𝑓�1 − ℎ𝑡𝑡𝑞𝑞 − (2 − 𝑓𝑓2)𝑣𝑣𝑛𝑛𝑡𝑡 − 𝑛𝑛𝑡𝑡𝜖𝜖𝑡𝑡𝑒𝑒�𝑤𝑤𝑡𝑡

𝑓𝑓

24

= 𝑑𝑑𝛼𝛼Φ𝜃𝜃𝑡𝑡𝑌𝑌𝑡𝑡 , and dividing both size by 𝐾𝐾𝑡𝑡, (A21) 𝐾𝐾𝑡𝑡+1𝐾𝐾𝑡𝑡

= 𝑑𝑑𝛼𝛼Φ𝜃𝜃𝑡𝑡𝑌𝑌𝑡𝑡𝐾𝐾𝑡𝑡

, and therefore (A22)

𝑌𝑌𝑡𝑡 = 𝑌𝑌�(𝐸𝐸𝑡𝑡𝑚𝑚𝑁𝑁𝑡𝑡

𝑚𝑚

𝐾𝐾𝑡𝑡)𝛼𝛼(𝐸𝐸𝑡𝑡

𝑓𝑓𝑁𝑁𝑡𝑡𝑓𝑓

𝐾𝐾𝑡𝑡)𝛼𝛼 (1 − 𝑓𝑓1ℎ𝑡𝑡

𝑞𝑞 − 𝑓𝑓2𝑣𝑣𝑛𝑛𝑡𝑡 − 𝑓𝑓3𝑛𝑛𝑡𝑡𝜖𝜖𝑡𝑡𝑒𝑒)𝛼𝛼 (1− ℎ𝑡𝑡𝑞𝑞 − (2 − 𝑓𝑓2)𝑣𝑣𝑛𝑛𝑡𝑡 − 𝑛𝑛𝑡𝑡𝜖𝜖𝑡𝑡𝑒𝑒)𝛼𝛼𝐾𝐾𝑡𝑡 ,

(A23)

𝑌𝑌𝑡𝑡𝐾𝐾𝑡𝑡

= 𝑌𝑌�( 1𝑘𝑘𝑡𝑡𝑚𝑚)𝛼𝛼( 1

𝑘𝑘𝑡𝑡𝑓𝑓)𝛼𝛼 (1 − 𝑓𝑓1ℎ𝑡𝑡

𝑞𝑞 − 𝑓𝑓2𝑣𝑣𝑛𝑛𝑡𝑡 − 𝑓𝑓3𝑛𝑛𝑡𝑡𝜖𝜖𝑡𝑡𝑒𝑒)𝛼𝛼 (1− ℎ𝑡𝑡𝑞𝑞 − (2 − 𝑓𝑓2)𝑣𝑣𝑛𝑛𝑡𝑡 − 𝑛𝑛𝑡𝑡𝜖𝜖𝑡𝑡𝑒𝑒)𝛼𝛼,

(A24)

where 𝑘𝑘𝑡𝑡𝑚𝑚 = 𝐾𝐾𝑡𝑡 𝐸𝐸𝑡𝑡𝑚𝑚𝑁𝑁𝑡𝑡

𝑚𝑚 and 𝑘𝑘𝑡𝑡𝑓𝑓 = 𝐾𝐾𝑡𝑡

𝐸𝐸𝑡𝑡𝑓𝑓𝑁𝑁𝑡𝑡

𝑓𝑓 .

Since 𝑒𝑒𝑡𝑡+1𝑓𝑓 = 𝑒𝑒𝑡𝑡+1𝑚𝑚 𝑘𝑘𝑡𝑡𝑚𝑚 = 𝑘𝑘𝑡𝑡

𝑓𝑓, (A25)

𝑌𝑌𝑡𝑡𝐾𝐾𝑡𝑡

= 𝑌𝑌� (1 − 𝑓𝑓1ℎ𝑡𝑡𝑞𝑞 − 𝑓𝑓2𝑣𝑣𝑛𝑛𝑡𝑡 − 𝑓𝑓3𝑛𝑛𝑡𝑡𝜖𝜖𝑡𝑡𝑒𝑒)𝛼𝛼 (1 − ℎ𝑡𝑡

𝑞𝑞 − (2 − 𝑓𝑓2)𝑣𝑣𝑛𝑛𝑡𝑡 − 𝑛𝑛𝑡𝑡𝜖𝜖𝑡𝑡𝑒𝑒)𝛼𝛼( 1

𝑘𝑘𝑡𝑡𝑓𝑓)2𝛼𝛼 .

(A26)

(iv) Dynamics for Education

From (6), (17) and (18),

𝑒𝑒𝑡𝑡+1𝑓𝑓 = 𝑒𝑒𝑡𝑡+1𝑚𝑚 = �̅�𝑒 � 𝜇𝜇𝐺𝐺𝑡𝑡

𝑛𝑛𝑡𝑡𝑎𝑎𝑁𝑁𝑡𝑡/2

�𝜈𝜈1

(𝑒𝑒𝑡𝑡𝑓𝑓)1−𝜈𝜈1[𝜖𝜖𝑡𝑡𝑒𝑒]𝜈𝜈2 = �̅�𝑒 �𝜇𝜇𝜇𝜇(1+𝑑𝑑)𝛼𝛼

𝑛𝑛𝑡𝑡𝑎𝑎/2

�𝜈𝜈1�(1−∅)−1𝑌𝑌𝑡𝑡

𝑁𝑁𝑡𝑡�𝜈𝜈1

(𝑒𝑒𝑡𝑡𝑓𝑓)1−𝜈𝜈1[𝜖𝜖𝑡𝑡𝑒𝑒]𝜈𝜈2

(A27)

By definition, 𝑌𝑌𝑡𝑡

0.5𝑒𝑒𝑡𝑡𝑓𝑓𝑁𝑁𝑡𝑡

= 𝑌𝑌𝑡𝑡𝐾𝐾𝑡𝑡

𝐾𝐾𝑡𝑡𝑒𝑒𝑡𝑡𝑓𝑓𝑁𝑁𝑡𝑡

𝑓𝑓 = 𝑌𝑌𝑡𝑡𝐾𝐾𝑡𝑡𝑘𝑘𝑡𝑡𝑓𝑓

= 𝑌𝑌� (1 − 𝑓𝑓1ℎ𝑡𝑡𝑞𝑞 − 𝑓𝑓2𝑣𝑣𝑛𝑛𝑡𝑡 − 𝑓𝑓3𝑛𝑛𝑡𝑡𝜖𝜖𝑡𝑡𝑒𝑒)𝛼𝛼 (1 − ℎ𝑡𝑡

𝑞𝑞 − (2 − 𝑓𝑓2)𝑣𝑣𝑛𝑛𝑡𝑡 − 𝑛𝑛𝑡𝑡𝜖𝜖𝑡𝑡𝑒𝑒)𝛼𝛼(𝑘𝑘𝑡𝑡𝑓𝑓)1−2𝛼𝛼 (A28)

(v) Dynamics for 𝒌𝒌𝒕𝒕𝒇𝒇

𝑘𝑘𝑡𝑡+1𝑓𝑓 = 𝐾𝐾𝑡𝑡+1

𝐸𝐸𝑡𝑡+1𝑓𝑓 𝑁𝑁𝑡𝑡+1

𝑓𝑓 = 𝐾𝐾𝑡𝑡+1𝐸𝐸𝑡𝑡+1𝑓𝑓 0.5𝑛𝑛𝑡𝑡𝑁𝑁𝑡𝑡/2

= 𝑑𝑑𝛼𝛼Φ𝜃𝜃𝑡𝑡𝑌𝑌𝑡𝑡0.25𝐸𝐸𝑡𝑡+1

𝑓𝑓 𝑛𝑛𝑡𝑡𝑁𝑁𝑡𝑡

= 𝑑𝑑𝛼𝛼Φ𝜃𝜃𝑡𝑡𝑌𝑌𝑡𝑡/𝑁𝑁𝑡𝑡

0.25𝑛𝑛𝑡𝑡�̅�𝑒�𝜇𝜇𝜇𝜇(1+𝑑𝑑)𝛼𝛼

𝑛𝑛𝑡𝑡/2 �𝜈𝜈1�𝑌𝑌𝑡𝑡𝑁𝑁𝑡𝑡

�𝜈𝜈1

(𝑒𝑒𝑡𝑡𝑓𝑓)1−𝜈𝜈1�𝜖𝜖𝑡𝑡

𝑒𝑒�𝜈𝜈2

= 𝑑𝑑𝛼𝛼Φ𝜃𝜃𝑡𝑡[2(1−𝑏𝑏)]𝜈𝜈2�̅�𝑒(𝑛𝑛𝑡𝑡)1−𝜈𝜈1

(𝜇𝜇𝜏𝜏(1 + 𝑑𝑑)𝛼𝛼)−𝜈𝜈1 � 𝑌𝑌𝑡𝑡0.5𝑒𝑒𝑡𝑡

𝑓𝑓𝑁𝑁𝑡𝑡�1−𝜈𝜈1

2(𝜖𝜖𝑡𝑡𝑒𝑒)−𝜈𝜈2

=Γ𝜃𝜃𝑡𝑡 � 𝑌𝑌𝑡𝑡0.5𝑒𝑒𝑡𝑡

𝑓𝑓𝑁𝑁𝑡𝑡�1−𝜈𝜈1

(𝜖𝜖𝑡𝑡𝑒𝑒)−𝜈𝜈2

= Γ𝜃𝜃𝑡𝑡 (𝑌𝑌�Γ1)1−𝜈𝜈1 (1− 𝑓𝑓ℎ𝑡𝑡𝑞𝑞)𝛼𝛼(1−𝜈𝜈1) (1 − ℎ𝑡𝑡

𝑞𝑞 − 𝑣𝑣𝑛𝑛𝑡𝑡 − 𝑛𝑛𝑡𝑡𝜖𝜖𝑡𝑡𝑒𝑒)𝛼𝛼(1−𝜈𝜈1)(𝜖𝜖𝑡𝑡𝑒𝑒)−𝜈𝜈2(𝑘𝑘𝑡𝑡𝑓𝑓)(1−2𝛼𝛼)(1−𝜈𝜈1)

25

(A29)

where Γ = 2𝑑𝑑𝛼𝛼Φ�̅�𝑒(𝑛𝑛𝑡𝑡)1−𝜈𝜈1

(𝜇𝜇𝜏𝜏(1 + 𝑑𝑑)𝛼𝛼)−𝜈𝜈1 .

(vi) Steady-State Growth Rate

From (A11), (A21), and (A24)

𝑌𝑌𝑡𝑡+1𝑁𝑁𝑡𝑡+1

=𝑌𝑌𝑡𝑡+1𝐾𝐾𝑡𝑡+1

𝐾𝐾𝑡𝑡+1𝑁𝑁𝑡𝑡+1

=𝑌𝑌𝑡𝑡+1𝐾𝐾𝑡𝑡+1

𝐾𝐾𝑡𝑡+11

𝑁𝑁𝑡𝑡+1

= 𝑌𝑌� (1 − 𝑓𝑓1ℎ𝑡𝑡𝑞𝑞 − 𝑓𝑓2𝑣𝑣𝑛𝑛𝑡𝑡 − 𝑓𝑓3𝑛𝑛𝑡𝑡𝜖𝜖𝑡𝑡𝑒𝑒)𝛼𝛼 (1 − ℎ𝑡𝑡

𝑞𝑞 − (2 − 𝑓𝑓2)𝑣𝑣𝑛𝑛𝑡𝑡

− 𝑛𝑛𝑡𝑡𝜖𝜖𝑡𝑡𝑒𝑒)𝛼𝛼(1

𝑘𝑘𝑡𝑡+1𝑓𝑓 )2𝛼𝛼 𝑑𝑑𝛼𝛼Φ𝜃𝜃𝑡𝑡𝑌𝑌𝑡𝑡

1

𝑛𝑛𝑡𝑡𝑁𝑁𝑡𝑡2

=2𝑌𝑌� (1 − 𝑓𝑓1ℎ𝑡𝑡𝑞𝑞 − 𝑓𝑓2𝑣𝑣𝑛𝑛𝑡𝑡 − 𝑓𝑓3𝑛𝑛𝑡𝑡𝜖𝜖𝑡𝑡𝑒𝑒)𝛼𝛼 (1 − ℎ𝑡𝑡

𝑞𝑞 − (2 − 𝑓𝑓2)𝑣𝑣𝑛𝑛𝑡𝑡 − 𝑛𝑛𝑡𝑡𝜖𝜖𝑡𝑡𝑒𝑒)𝛼𝛼( 1

𝑘𝑘𝑡𝑡+1𝑓𝑓 )2𝛼𝛼 𝑑𝑑𝛼𝛼Φ𝜃𝜃𝑡𝑡

1𝑛𝑛𝑡𝑡

𝑌𝑌𝑡𝑡𝑁𝑁𝑡𝑡

In the steady state, we have

1 + 𝛾𝛾𝑌𝑌/𝑁𝑁 = 2𝑌𝑌�(1 − 𝑓𝑓ℎ𝑞𝑞∗ − 𝑓𝑓2𝑣𝑣𝑛𝑛∗ − 𝑓𝑓3𝑛𝑛∗𝜖𝜖𝑒𝑒∗)𝛼𝛼(1 − ℎ𝑞𝑞∗ − (2 − 𝑓𝑓2)𝑣𝑣𝑛𝑛∗ −𝑛𝑛∗𝜖𝜖𝑒𝑒∗)𝛼𝛼(𝑘𝑘∗)−2𝛼𝛼𝑑𝑑𝛼𝛼Φ𝜃𝜃∗(𝑛𝑛∗)−1

(A30)

where the variables with * are steady state values and 𝑘𝑘𝑓𝑓∗ = ( 𝐾𝐾𝑒𝑒𝑓𝑓𝑁𝑁𝑓𝑓

)∗.

When 𝑓𝑓 increases, depending on what happens to the steady state solutions, particularly ℎ𝑞𝑞, the steady state growth rate can either increase or not.

26

Figure 1. Cohort Employment Rate in 2013: Male vs. Female in Korea

Source: OECD (http://stats.oecd.org)

0

10

20

30

40

50

60

70

80

90

100

15~19 20~24 25~29 30~34 35~39 40~44 45~49 50~54 55~59 60~64

Male

Female

27

Figure 2. The Impact of Gender Equality Policies 2.1 Fertility

11.

11.

21.

31.

4

0 1 2 3 4 5Time

(v from 2.8099 to 2.5)Lower time cost for child rearing

11.

11.

2

0 1 2 3 4 5Time

(d from 0.6 to 0.7)Lower discrimination in the labor market

1.1

1.15

1.2

0 1 2 3 4 5Time

(f2 from 0.3279 to 0.6667)Increase the time spent by a male on child rearing

28

2.2 Female’s Time Allocated to Market Production

54.3

54.4

54.5

0 1 2 3 4 5Time

(v from 2.8099 to 2.5)Lower time cost for child rearing

5454

.555

55.5

56

0 1 2 3 4 5Time

(d from 0.6 to 0.7)Lower discrimination in the labor market

5454

.555

55.5

56

0 1 2 3 4 5Time

(f2 from 0.3279 to 0.6667)Increase the time spent by a male on child rearing

29

2.3 Per Capita Output Growth

3.2

3.3

3.4

3.5

3.6

3.7

0 1 2 3 4 5Time

(v from 2.8099 to 2.5)Lower time cost for child rearing

3.5

3.6

3.7

3.8

3.9

0 1 2 3 4 5Time

(d from 0.6 to 0.7)Lower discrimination in the labor market

3.55

3.6

3.65

3.7

0 1 2 3 4 5Time

(f2 from 0.3279 to 0.6667)Increase the time spent by a male on child rearing

30

Table 1. Various Measures of Gender Gap in Korea Women Men

Employment ratea Master’s degrees or higher 59.4% 80.6% Bachelor degrees 50.0% 58.7%

Distribution of workers by employment typesb Employer 3.2% 8.4% Own account workers 12.4% 20.3% Unpaid family workers 10.7% 1.2% Regular 37.1% 49.5% Temporary 28.7% 15.0% Daily 7.9% 6.8%

Relative average monthly wagec Overall 64% 100% Bachelor degrees or higher 66% 100% Professional occupations (doctors, lawyers, etc.) 62% 100%

Representation in politics and government 18th National Assembly members 41 258 Federal government employeesd 41% 59% Members of corporate boardse 1.9% [12%] 98.1% [88%]

Data sources: a. Ministry of Education, Science and Technology. Statistical Yearbook of Education, 2011. b. Statistics Korea. Annual Report on the Economically Active Population Survey, 2011. c. Ministry of Employment and Labor. Survey on Labor Conditions by Type of Employment, 2010. d. Statistics Korea. Annual Report on the Economically Active Population Survey, 2011. e. GMI Ratings. “GMI Ratings’ 2013 Women On Board Survey”, April 2013. Note: 1. Average monthly wage is based on the total of the monthly salary and the monthly share of the annual bonus. 2. Members of corporate boards comprise both private and public firms.

31

Table 2. The Global Gender Gap Index Ranking in 2013

Country Overall Economic participation and opportunity

Educational attainment

Health and Survival

Political empowerment

Iceland 1 22 1 97 1 Finland 2 19 1 1 2 Norway 3 1 1 93 3 Sweden 4 14 38 69 4 Philippines 5 16 1 1 10 Germany 14 46 86 49 15 United States 23 6 1 33 60 Sri Lanka 55 109 48 1 30 Singapore 58 12 105 85 90 Thailand 65 50 78 1 89 China 69 62 81 133 59 Vietnam 73 52 95 132 80 Bangladesh 75 121 115 124 7 Indonesia 95 103 101 107 75 India 101 124 120 135 9 Malaysia 102 100 73 75 121 Japan 105 104 91 34 118 Korea, Rep. 111 118 100 75 86 Pakistan 135 135 129 124 64 Source: Bekhouch, Y., Hausmann, R., Tyson, L. D., & Zahidi, S. (2013, September). The Global Gender Gap Report 2013.

32

Table 3. Calibrated Parameters

Parameter Value Description Households

ρ 0.5982* Time discount rate (over 30 years)

σ 0.8 Inverse of elasticity of substitution

𝑃𝑃𝐴𝐴 0.987 Survival probability

δ 1.05 Preference parameter for number of children

𝜂𝜂𝑒𝑒 0.2 Preference parameters for children’s education 𝜂𝜂𝑞𝑞 12 Family preference parameter for home

production output 𝜂𝜂𝑐𝑐 3.5 Preference parameter for consumption 𝑣𝑣 2.8099*

Home output

γ 0.122 Curvature of production function

f1 0.5897* Bargaining power of a female in home production

f2 0.3279* Bargaining power of a female in child rearing f3 0.3077* Bargaining power of a female in child education 𝑞𝑞� 23.6313* 𝜒𝜒 0.8

Market output

α 0.4 Elasticity with respect to (wrt) labor input

d 0.6 Gender bias in the workplace 𝑌𝑌� 1

Human capital

𝜐𝜐1 0.4 Elasticity wrt public spending in education 𝜐𝜐2 0.3 Elasticity wrt public-private ratio

�̅�𝑒 4.2797*

Government

τ 0.163 Tax rate on marketed output

μ 0.39 Education spending efficiency parameter

∅ 3 Factor of unproductive, exogenous government expenditure to educational expenditure

Note: Parameter values with * are derived from our own calibration of the model, using the World Development Indicators by the World Bank, data from the Bank of Korea, and Korea Time Use Survey data. The rest are from the macroeconomic literature and Kim et al. (2016). See section 2.2 for details.

33

Table 4. Steady State Solutions

Variables Value Description

𝑝𝑝𝑐𝑐𝑛𝑛 1.17 Fertility rate (𝑛𝑛 = 1.17)

ℎ𝑚𝑚 0.7592 Labor force participation rate of males

ℎ𝑤𝑤 0.5443 Labor force participation rate of females

𝜃𝜃 0.1610 Net private savings rate

𝛾𝛾𝑌𝑌/𝑁𝑁 1.889 Per capita growth rate (= 1.036030 – 1)

Source: Authors’ calculation

34

Table 5. Steady-state Values for the Hypothetical Cases with Gender Equality

Fertility Female labor participation rate (%)

Per capita output growth rate

(i) Current level 1.17 54.43 0.0360

Complete gender equality

(ii) d =1, f1 = f2 = f3 = 1

0.98

67.51

0.0406

Partial equality

(iii) d = 1

0.97

59.30

0.0434

(iv) f1 = 1 1.29 62.24 0.0346

(v) f2 = 1 1.10 56.50 0.0378

(vi) f3 = 1 1.08 55.43 0.0345

(vii) d =1, f1 = 1 1.04 64.84 0.0420

Note: See Table 3 for the definition of the parameters, d, f1, f2, and f3.

Source: Authors’ calculation